MathProblem 72 A jar and an amoeba problem

A jar begins with one amoeba. Every minute, every amoeba turns into 0, 1, 2, or 3 amoebae with probability 25% for each case ( dies, does nothing, splits into 2, or splits into 3). What is the probability that the amoeba population eventually dies out?

Solution

假设这个概率为 \(p\). 那么对于 \(n\) 个，它们都是相互独立的，所以这种情况下为 \(p^n\).

因此:

\[p=\frac{1}{4}(1+p+p^2+p^3) \]

得到：

\[p^3+p^2-3p+1=0\Rightarrow (p-1)(p^2+2p-1)=0 \]

求解即可